Calculate the following integral using Residue Theorem

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- Jul 9th 2009, 11:04 PMRichmondResidue Theorem
Calculate the following integral using Residue Theorem

from to - Jul 9th 2009, 11:42 PMMoo
Hello,

Let

Obviously, i and -i are double poles, because

So, using the general formula for a pole of order n :

we easily get the residue of f at i (the one that will be useful for the computation) :

which is easy to compute.

__Spoiler__:

Now, use the semicircle contour :

http://upload.wikimedia.org/wikipedi..._residus_2.png

And keep the poles with positive imaginary part. So only i.

And then - Jul 10th 2009, 12:22 AMRichmond
ok. so for

and

Am I right to say that you took semi contour and the positive side of the imaginary because of the -infinity to infinity?

The last part would be

- Jul 10th 2009, 12:32 AMMoo
Hey...

So

You got wrong for the second one too.

Isn't it surprising to get a negative answer for the integral of a positive function ?

Quote:

Am I right to say that you took semi contour and the positive side of the imaginary because of the -infinity to infinity?

But the result is actually the same since

But more commonly, we take the upper semi circle ;)

I hope I'm clear :s - Jul 10th 2009, 12:36 AMRichmond
Ok. thanks for the clearance..

Yeap. I got a careless calculation error there, thanks for pointing that out.